Consider a strong explosion (such as nuclear bombs) that releases a large amount of energy E in a small volume during a short time interval. This will create a strong spherical
shock wave propagating outwards from the explosion center. The self-similar solution tries to describe the flow when the shock wave has moved through a distance that is extremely large when compared to the size of the explosive. At these large distances, the information about the size and duration of the explosion will be forgotten; only the energy released E will have influence on how the shock wave evolves. To a very high degree of accuracy, then it can be assumed that the explosion occurred at a point (say the origin r=0) instantaneously at time t=0. The shock wave in the self-similar region is assumed to be still very strong such that the pressure behind the shock wave p_1 is very large in comparison with the pressure (
atmospheric pressure) in front of the shock wave p_0, which can be neglected from the analysis. Although the pressure of the undisturbed gas is negligible, the density of the undisturbed gas \rho_0 cannot be neglected since the density jump across strong shock waves is finite as a direct consequence of
Rankine–Hugoniot conditions. This approximation is equivalent to setting p_0=0 and the corresponding sound speed c_0=0, but keeping its density non zero, i.e., \rho_0\neq 0. The only parameters available at our disposal are the energy E and the undisturbed gas density \rho_0. The properties behind the shock wave such as p_1,\,\rho_1 are derivable from those in front of the shock wave. The only non-dimensional combination available from r,\,t,\,\rho_0 and E is r\left(\frac{\rho_0}{Et^2}\right)^{1/5}. It is reasonable to assume that the evolution in r and t of the shock wave depends only on the above variable. This means that the shock wave location r=R(t) itself will correspond to a particular value, say \beta, of this variable, i.e., R= \beta \left(\frac{Et^2}{\rho_0}\right)^{1/5}. The detailed analysis that follows will, at the end, reveal that the factor \beta is quite close to unity, thereby demonstrating (for this problem) the quantitative predictive capability of the
dimensional analysis in determining the shock-wave location as a function of time. The propagation velocity of the shock wave is D = \frac{\mathrm{d}R}{\mathrm{d}t} = \frac{2R}{5t} = \frac{2\beta}{5} \left(\frac{E}{\rho_0t^3}\right)^{1/5} With the approximation described above,
Rankine–Hugoniot conditions determines the gas velocity immediately behind the shock front v_1, p_1 and \rho_1 for an
ideal gas as follows \begin{align} v_1 &= \frac{2 D}{\gamma+1}, & p_1 &= \frac{2 \rho_0 D^2}{\gamma+1}, & \rho_1 &= \rho_0 \frac{\gamma+1}{\gamma-1} \end{align} where \gamma is the
specific heat ratio. Since \rho_0 is a constant, the density immediately behind the shock wave is not changing with time, whereas v_1 and p_1 decrease as t^{-3/5} and t^{-6/5}, respectively.
Self-similar solution The gas motion behind the shock wave is governed by
Euler equations. For an ideal
polytropic gas with spherical symmetry, the equations for the fluid variables such as radial velocity v(r,t), density \rho(r,t) and pressure p(r,t) are given by \begin{align} \frac{\partial v}{\partial t} + v \frac{\partial v}{\partial r} + \frac{1}{\rho}\frac{\partial p}{\partial r} &= 0, \\[1ex] \frac{\partial \rho}{\partial t} + \frac{\partial \left(\rho v\right)}{\partial r} + \frac{2\rho v}{r} &= 0, \\[1ex] \frac{\partial }{\partial t} \ln \frac{p}{\rho^\gamma}+ v\frac{\partial }{\partial r} \ln \frac{p}{\rho^\gamma} &=0. \end{align} At r=R(t), the solutions should approach the values given by the Rankine–Hugoniot conditions defined in the previous section. The variable pressure can be replaced by the sound speed c(r,t) since pressure can be obtained from the formula c^2=\gamma p/\rho. The following non-dimensional self-similar variables are introduced, \begin{align} \xi &= \frac{r}{R(t)}, & V(\xi) &= \frac{5tv}{2r}, \\ G(\xi) &= \frac{\rho}{\rho_0}, & Z(\xi) &= \left(\frac{5 t c}{2r}\right)^2. \end{align} The conditions at the shock front \xi=1 become \begin{align} V(1) &= \frac{2}{\gamma+1}, & G(1) &= \frac{\gamma+1}{\gamma-1}, & Z(1) &= \frac{2\gamma(\gamma-1)}{(\gamma+1)^2}. \end{align} Substituting the self-similar variables into the governing equations will lead to three ordinary differential equations. Solving these differential equations analytically is laborious, as shown by Sedov in 1946 and von Neumann in 1947. G. I. Taylor integrated these equations numerically to obtain desired results. The relation between Z and V can be deduced directly from energy conservation. Since the energy associated with the undisturbed gas is neglected by setting the total energy of the gas within the shock sphere must be equal to E. Due to
self-similarity, it is clear that not only the total energy within a sphere of radius \xi=1 is constant, but also the total energy within a sphere of any radius \xi (in dimensional form, it says that total energy within a sphere of radius r that moves outwards with a velocity v_n=2r/5t must be constant). The amount of energy that leaves the sphere of radius r in time dt due to the gas velocity v is {{nowrap|4\pi r^2\rho v(h+v^2/2)\mathrm{d}t,}} where h is the
specific enthalpy of the gas. In that time, the radius of the sphere increases with the velocity v_n and the energy of the gas in this extra increased volume is {{nowrap|4\pi r^2 \rho v_n(e+v^2/2)\mathrm{d}t,}} where e is the
specific energy of the gas. Equating these expressions and substituting e = c^2 / \gamma(\gamma-1) and h = c^2/(\gamma-1) that is valid for ideal polytropic gas leads to Z = \frac{\gamma(\gamma-1)(1-V)V^2}{2(\gamma V-1)}. The continuity and energy equation reduce to \begin{align} \frac{\mathrm{d} V }{\mathrm{d}\ln \xi} - (1-V) \frac{\mathrm{d}\ln G}{\mathrm{d}\ln\xi} &= - 3V \\[1ex] \frac{\mathrm{d}\ln Z}{\mathrm{d}\ln\xi} - (\gamma-1) \frac{\mathrm{d}\ln G}{\mathrm{d}\ln\xi} &= -\frac{5-2V}{1-V}. \end{align} Expressing \mathrm{d}V/\mathrm{d}\ln\xi and \mathrm{d}\ln G/\mathrm{d}V as a function of V only using the relation obtained earlier and integrating once yields the solution in implicit form, \begin{align} \xi^5 &= \left[\frac{1}{2}(\gamma+1)V\right]^{-2} \left\{\frac{\gamma+1}{7-\gamma}[5-(3\gamma-1)V]\right\}^{\nu_1}\left[\frac{\gamma+1}{\gamma-1}(\gamma V-1)\right]^{\nu_2}, \\[1ex] G &= \frac{\gamma+1}{\gamma-1}\left[\frac{\gamma+1}{\gamma-1}(\gamma V-1)\right]^{\nu_3} \left\{\frac{\gamma+1}{7-\gamma}[5-(3\gamma-1)V]\right\}^{\nu_4}\left[\frac{\gamma+1}{\gamma-1}(1-V)\right]^{\nu_5} \end{align} where \begin{align} \nu_1 &= -\frac{13\gamma^2-7\gamma+12}{(3\gamma-1)(2\gamma+1)}, & \nu_2 &= \frac{5(\gamma-1)}{2\gamma+1}, \\[1ex] \nu_3 &= \frac{3}{2\gamma+1}, & \nu_4 &= -\frac{\nu_1}{2-\gamma}, & \nu_5 &= - \frac{2}{2-\gamma}. \end{align} The constant \beta that determines the shock location can be determined from the conservation of energy E = \int_0^R \rho\left[\frac{v^2}{2} + \frac{c^2}{\gamma(\gamma-1)}\right] 4 \pi r^2\mathrm{d}r to obtain \beta^5 \frac{16\pi}{25} \int_0^1 G\left[\frac{V^2}{2} + \frac{Z}{\gamma(\gamma-1)}\right] \xi^4 \,\mathrm{d}\xi = 1. For air, \gamma = 7/5 and The solution for \gamma = 7/5 is shown in the figure by graphing the curves of and where T is the temperature.
Asymptotic behavior near the central region The asymptotic behavior of the central region can be investigated by taking the limit \xi\to 0. From the figure, it can be observed that the density falls to zero very rapidly behind the shock wave. The entire mass of the gas which was initially spread out uniformly in a sphere of radius R is now contained in a thin layer behind the shock wave, that is to say, all the mass is driven outwards by the acceleration imparted by the shock wave. Thus, most of the region is basically empty. The pressure ratio also drops rapidly to attain the constant value p_c. The temperature ratio follows from the
ideal gas law; since
density ratio decays to zero and the pressure ratio is constant, the temperature ratio must become infinite. The limiting form for the density is given as follows \frac{\rho}{\rho_1} \sim \xi^{3/(\gamma-1)}, \quad \frac{p}{p_1} \to p_c, \quad \frac{T}{T_1} \sim \xi^{-3/(\gamma-1)} \quad \text{as} \quad \xi\to 0. Remember that the density \rho_1 is time-independent whereas p_1\sim t^{-6/5} which means that the actual pressure is in fact time dependent. It becomes clear if the above forms are rewritten in dimensional units, \rho \sim r^{3/(\gamma-1)}t^{-6/5(\gamma-1)}, \quad p\to p_c t^{-6/5}, \quad T \sim r^{-3/(\gamma-1)}t^{(6/5)(2-\gamma)/(\gamma-1)} \quad \text{as} \quad r \to 0. The velocity ratio has the linear behavior in the central region, \frac{v}{v_1} \sim \xi \quad \text{as} \quad \xi \to 0 whereas the behavior of the velocity itself is given by v \sim r t^{1/5} \quad \text{as} \quad r \to 0. ==Final stage of the blast wave==